Physics (Part 1 - Part 3)

422 CHAPTER 12 | The Laws of Thermodynamics
Tip 12.5 Don’t Confuse
the W’s
The symbol W used here is a probability,
not to be confused with the
same symbol used for work.
APPLICATION
The Direction of Time
In light of this new view of entropy, Boltzmann found another method for calculating
entropy through use of the relation
S 5 k B
ln W [12.18]
where k B
5 1.38 3 10 223 J/K is Boltzmann’s constant and W is a number proportional
to the probability that the system has a particular configuration. The symbol
“ln” again stands for natural logarithm, discussed in Appendix A.
Equation 12.18 could be applied to a bag of marbles. Imagine that you have
100 marbles—50 red and 50 green—stored in a bag. You are allowed to draw four
marbles from the bag according to the following rules: Draw one marble, record
its color, return it to the bag, and draw again. Continue this process until four
marbles have been drawn. Note that because each marble is returned to the bag
before the next one is drawn, the probability of drawing a red marble is always the
same as the probability of drawing a green one.
The results of all possible drawing sequences are shown in Table 12.3. For example,
the result RRGR means that a red marble was drawn first, a red one second,
a green one third, and a red one fourth. The table indicates that there is only one
possible way to draw four red marbles. There are four possible sequences that produce
one green and three red marbles, six sequences that produce two green and
two red, four sequences that produce three green and one red, and one sequence
that produces all green. From Equation 12.18, we see that the state with the greatest
disorder (two red and two green marbles) has the highest entropy because it is
most probable. In contrast, the most ordered states (all red marbles and all green
marbles) are least likely to occur and are states of lowest entropy.
The outcome of the draw can range between these highly ordered (lowestentropy)
and highly disordered (highest-entropy) states. Entropy can be regarded
as an index of how far a system has progressed from an ordered to a disordered
state.
The second law of thermodynamics is really a statement of what is most probable
rather than of what must be. Imagine placing an ice cube in contact with a hot
piece of pizza. There is nothing in nature that absolutely forbids the transfer of
energy by heat from the ice to the much warmer pizza. Statistically, it’s possible for
a slow-moving molecule in the ice to collide with a faster-moving molecule in the
pizza so that the slow one transfers some of its energy to the faster one. When the
great number of molecules present in the ice and pizza are considered, however,
the odds are overwhelmingly in favor of the transfer of energy from the fastermoving
molecules to the slower-moving molecules. Furthermore, this example
demonstrates that a system naturally tends to move from a state of order to a state
of disorder. The initial state, in which all the pizza molecules have high kinetic
energy and all the ice molecules have lower kinetic energy, is much more ordered
than the final state after energy transfer has taken place and the ice has melted.
Even more generally, the second law of thermodynamics defines the direction
of time for all events as the direction in which the entropy of the universe
increases. Although conservation of energy isn’t violated if energy flows spontaneously
from a cold object (the ice cube) to a hot object (the pizza slice), that event
Table 12.3 Possible Results of Drawing Four Marbles from a Bag
Total Number
End Result Possible Draws of Same Results
All R RRRR 1
1G, 3R RRRG, RRGR, RGRR, GRRR 4
2G, 2R RRGG, RGRG, GRRG, RGGR, GRGR, GGRR 6
3G, 1R GGGR, GGRG, GRGG, RGGG 4
All G GGGG 1
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